Given a nonlinear control system d/dt x=f(x,u), a theorem due to R.W. Brockett,  reported in a now famous paper published in 1983, points out a necessary condition for the existence of differentiable pure-state feedbacks capable of stabilizing equilibria of  this nonlinear system asymptotically. The Transverse Function control approach addresses a shortcoming of current Nonlinear Feedback Control Theory concerning the class of locally controllable critical systems (i.e. the linearization of which at an equilibrium point of interest is not controllable) that do not satisfy Brockett’s necessary condition. This class of systems is well populated since it contains, in particular, all nonholonomic mechanisms and common ground vehicles (wheeled mobile robots, cars, trucks,…), as well as many underactuated systems (ships, hovercrafts, submarines, satellites subjected to actuation failure,…). All controllers currently implemented on these systems fail to perform properly when the objective is the asymptotic stabilization of points, or trajectories, where the sytem is critical. Nor can they deal with reference trajectories that do not belong to the set of solutions of the system (non-feasible trajectories).
The core of the approach relies on a theorem that roughly states the following: given a n-dimensional driftless control system d/dt x=∑_i=1^m X_i(x)u_i, with m<n, such that the vector fields X_i satisfy the so-called Lie Algebra Rank Property (LARC) in a neighbourhood of x=0 –a property which implies that the system is locally controllable at x=0–, there exist differentiable functions f(α) defined on a p-dimensional compact manifold and taking values in a ball (arbitrarily small) centered on x=0 such that the matrix H(α)=[X₁(f(α)),…,X_m(f(α)),∂f/∂α₁(α),…,∂f/∂α_p(α)] is of rank n, ∀α. In other words, the system’s control vector fields X_i evaluated at f(α), complemented with the differential of the function f, span the entire state space for all values of the variables on which the function f depends. It is in this sense that this function is transversal (or transverse) to the system’s vector fields. How is this transversality property useful to control the system? Calculating the time-derivative of the “modified” state z=x-f(α) yields d/dt z=H(α)ũ+o(z,u), with ũ the “extended” control vector composed of u and -d/dt α, and o(z,u) a vector of higher order terms near zero. Exponential stabilization of z=0 is then a very simple matter. One can take, for instance, ũ=-k H(α)^† z, with H(α)^† denoting a pseudo-inverse of H(α) and k a positive number. This yields the closed-loop equation d/dt z=-kz+o(z,u), from which (local) exponential stability of z=0 follows directly. Therefore, if f(α) is small, the original state x converges to a small neighborhood of zero. The theorem that we have proved is in fact slightly more general, and more elegantly expressed when considering a system which is left-invariant on a Lie group. In this latter case, it is possible to obtain a global asymptotic stabilization result for the modified state vector, defined in this case as          z=x• f(α)^-1, with • denoting the group operation and f(α)^-1 the group inverse of f(α).
Also the proof of the theorem is constructive and it provides a systematic way of computing explicit transverse functions. To summarize, the use of transverse functions, the existence of which is equivalent to the property of local controllability for driftless systems, allows for a simple way to achieve the practical stabilization of x=0, meaning that any arbitrarily small neighborhood of this point can be rendered uniformly exponentially stable. Moreover, by considering a reference trajectory x_r(t) on the Lie group on which the system is left-invariant, defining the tracking error x_err=x_r^-1•x, and setting z= x_err • f(α)^-1, one can (practically) stabilize this trajectory, whether it is, or is not, a solution to the system (i.e. it does not have to be feasible).